Final answer:
To find a basis for R4 containing v=(1,-1,1,-1) and w=(0,1,0,1), we can use the spanning set method. A basis for R4 will consist of linearly independent vectors that span the space.
Step-by-step explanation:
To find a basis for R4 containing v=(1,-1,1,-1) and w=(0,1,0,1), we can use the spanning set method. A basis for R4 will consist of linearly independent vectors that span the space.
Step 1: Determine if v and w are linearly independent. We can check this by setting up the equation a(1,-1,1,-1) + b(0,1,0,1) = (0,0,0,0), where a and b are scalar values. This gives us the system of equations:
a = 0 and -a + b = 0. From the first equation, we get a = 0. Substituting into the second equation, we get -0 + b = 0, which gives us b = 0. Therefore, v and w are linearly independent.
Step 2: Since v and w are linearly independent, they form a basis for R4. Therefore, a basis for R4 that contains v and w is {v, w}.